MathDB

2

Part of 2021 Iranian Geometry Olympiad

Problems(3)

diagonal of inscribed KLMN in square ABCD is // to side of ABCD having 1/2 area

Source: Iranian Geometry Olympiad 2021 IGO Elementary p2

1/25/2022
Points K,L,M,NK, L, M, N lie on the sides AB,BC,CD,DAAB, BC, CD, DA of a square ABCDABCD, respectively, such that the area of KLMNKLMN is equal to one half of the area of ABCDABCD. Prove that some diagonal of KLMNKLMN is parallel to some side of ABCDABCD.
Proposed by Josef Tkadlec - Czech Republic
inscribedgeometryareassquare
IGO 2021 P2

Source: intermediate p2

12/30/2021
Let ABCDABCD be a parallelogram. Points E,FE, F lie on the sides AB,CDAB, CD respectively, such that ∠EDC=∠FBC\angle EDC = \angle FBC and ∠ECD=∠FAD\angle ECD = \angle FAD. Prove that AB≥2BCAB \geq 2BC.
Proposed by Pouria Mahmoudkhan Shirazi - Iran
IGOgeometry
two circles intersecting, prove that three lines are concurrent

Source: IGO 2021 Advanced P2

12/30/2021
Two circles Γ1\Gamma_1 and Γ2\Gamma_2 meet at two distinct points AA and BB. A line passing through AA meets Γ1\Gamma_1 and Γ2\Gamma_2 again at CC and DD respectively, such that AA lies between CC and DD. The tangent at AA to Γ2\Gamma_2 meets Γ1\Gamma_1 again at EE. Let FF be a point on Γ2\Gamma_2 such that FF and AA lie on different sides of BDBD, and 2∠AFC=∠ABC2\angle AFC=\angle ABC. Prove that the tangent at FF to Γ2\Gamma_2, and lines BDBD and CECE are concurrent.
geometryIGOconcurrent